Chapter 7

CHAPTER 7

Friction Loss in Flow through Granular Media

159

Chapter Objectives

1. Quantify friction loss in fluid flow through granular media.2. Estimate minimum fluidization velocity for packed beds.

Granular Media Used in Treatment Systems

Liquid flow through solid granular media is a common process in many treatmentsystems. Granular media are used in sand and mixed media filtration, granularactivated carbon contactors, and ion exchange processes. As the fluid must movein the stationary pores of the media, energy is lost from skin friction (fluid flow-ing past granular surfaces) and form friction (fluid flowing around grains). Engi-neers must be able to predict the frictional loss to properly size the pumps andpiping servicing the process employing granular media.

Granular media processes may be contained by closed vessels, as shown bythe ion exchange process in Fig. 7-1, or open vessels, as for the slow sand filterin Fig. 7-2. In an open vessel, the surface of the media or the fluid above themedia is open to the atmosphere. In a closed vessel, the difference in pressurebetween the inlet and discharge sides of the media must be accounted for in thesystem design, or the desired flow rate may not be attainable. In the open vessel,the desired flow must be achievable with the head of fluid above the media in adownflow configuration.

Granular Media Filtration

Filtration is the removal of suspended solids and colloidally sized particles (parti-cles with one dimension between 1 nm and 1 �m) by passing through a mediasuch as sand. The particles are destabilized and will attach to the media surfacesin the filter if the conditions are amenable to attachment. See Fig. 7-3. Theremoval of particles from the liquid occurs within the media bed, but the flowthrough the filter produces friction loss as well. As solids are removed from the

160 TREATMENT SYSTEM HYDRAULICS

Figure 7-1. Closed-vessel ion-exchange process.

Figure 7-2. Slow sand filter, open-vessel filter process. Above the media isopen to atmospheric pressure.

fluid stream, the pore volume decreases and the frictional loss increases above theclean bed friction loss, as illustrated in Fig. 7-4. As available surface area becomesoccupied by attached solids during a filter run, the quality of the effluent decreases.Types of granular media filters include the following:

• Rapid, granular media filters: These filters may be single media (sand), dualmedia (sand and anthracite), or triple media (sand, anthracite, and garnet).Filtration rates usually range between 80 and 400 L/m2�min (�2 to 10 gpm/ft2)(Tchobanoglous and Schroeder 1985), with media depths less than 1 m (�3 ft).The sand filter must be backwashed when the effluent no longer meets accept-able quality levels, or when the frictional loss is too high for the system.

• Slow sand filters: Slow sand filters have a much lower filtration rate than rapidsand filters, with filtration rates ranging from 2 to 5 L/m2�min (�0.05 to 0.12gpm/ft2) (Tchobanoglous and Schroeder 1985). In these filters, suspendedsolids and colloids are removed in a surface mat, called the “schmutzdecke”.When the slow sand filter becomes clogged (manifested by high head loss), theupper layer is removed and replaced with clean sand.

Granular Activated Carbon Contactors

Granular activated carbon (GAC) is the granular form of activated carbon, whichis produced by heating an organic such as hardwood, coal, or nut shells without

FRICTION LOSS IN FLOW THROUGH GRANULAR MEDIA 161

particles inaqueous solution

particle attachment

granular media surface

fluid flow

Figure 7-3. Attachment during granular media filtration process.

oxygen present (pyrolysis). The carbon has a high surface area, on the order of1,000 m2/g. Organic contaminants (e.g., benzene and toluene) have an affinity forthe activated carbon and so are removed from the fluid stream as it passes throughthe activated carbon. Granular activated carbon may be used as fixed beds inpacked bed, closed vessel contactors similar to that shown in Fig. 7-1. It can also beused in open vessel contactors. As the fluid passes through the GAC bed, it losesenergy from frictional loss.

It is common for more than one fixed bed contactor to be operated in series,one after another, so that the effluent quality from the series can be maintained.When breakthrough of contaminant occurs in the effluent from the first contac-tor, it can be taken offline, and the GAC changed (regenerated or disposed of).When two contactors in series are used in this fashion, the second contactor pre-vents contaminants from being discharged because the second contactor’s carbonstill effectively removes contaminants after the first contactor’s carbon is saturatedwith contaminants. See Fig. 7-5.

Friction Loss in Granular Media

Fluids flowing through beds of granular media will lose energy. When designinga system using a granular media process, it is important to be able to predict the


filter run time

filte

r fr

ictio

n lo

ss, e

fflue

nt tu

rbid

ity

friction loss

effluent turbidity

Figure 7-4. Normal operation of a sand filter showing increasing frictionloss with run time and increase in effluent turbidity.

frictional loss for flow through these packed beds. The following presents a deri-vation for an equation that may be used to calculate friction loss for fluids flowingthrough packed beds.

Characteristics of Fluid Flow in Granular Media

For a packed bed of granular media, the media grains are supported on oneanother, and the fluid must flow around the grains to pass through the bed. Thefluid cannot flow through the solid, so the solid portion of the bed (made up ofthe grains) is precluded from fluid flow. The void volume fraction available forfluid flow, defined as porosity ε, is

(7-1)

Typical porosity values are on the order of 0.3 to 0.4 dependent on the grainsize, shape, and packing. The presence of the solid grains in the contactorincreases the fluid velocity through the contactor relative to the velocity in a con-tactor that is empty because the cross-sectional area for flow is reduced by the solid

ε �volume of voids in bed

total bed volume


Figure 7-5. Series configuration fixed-bed granular activated carboncontactors in operation.

grains. The empty bed velocity V0 is also called the superficial velocity. The veloc-ity in the pores that the fluid flows through is

(7-2)

For fluid flow through granular media, we can characterize the flow with theReynolds number as was done for conduit flow. The Reynolds number in generalterms is written for a characteristic length, characteristic velocity, fluid density, andfluid viscosity as

(7-3)

For flow through a packed bed, it is common, but not universal, to choose thegrain diameter d as the characteristic length and the fluid superficial velocity V0 asthe characteristic velocity, so that

(7-4)

where � is the fluid density and � is the absolute fluid viscosity.The value of the Reynolds number, as for flow through a pipe, provides an

indication of the type of flow in a packed bed. For flow through granular mediawith low Reynolds numbers, � V 1, laminar flow is expected. For low-Reynolds-number flow, viscous effects dominate and there is little lateral mixing during flowthrough the media. There are no cross-currents or eddies. For flow with highReynolds numbers, � W 1, turbulent flow is expected as viscous effects are negli-gible. Random fluid motion predominates in that eddies form and lateral mixingoccurs.

Derivation of Friction Loss Equation

Consider a cylindrical element with the axis running parallel to flow in a packedbed as illustrated in Fig. 7-6. For this derivation, assume that steady-state condi-tions exist and that the fluid is incompressible (constant �). A fluid flowing throughthe cylinder will exert a force on the solid surfaces because of wall and form drag.That is, the fluid will lose energy. We can define a dimensionless friction factor e,such that the force exerted on the surfaces by the fluid is a function of the kineticenergy per unit volume of the fluid:

FK � A � K � e (7-5)

where A is the wetted surface area of the cylinder, K is the kinetic energy of thefluid per unit volume (1⁄2��Vpore

2) and e is a dimensionless friction factor. Con-ducting a force balance on the fluid in the element, as shown in Fig. 7-7, we have

F0 � FK � FL � 0 (7-6)

� �d V� �0 �

�

� �characteristic length characteristic velo� ccity fluid density

absolute fluid viscosity

�

VV

pore � 0

ε


where F0 is the force on the upstream face of the cylinder due to the upstream pres-sure and FL is the force on the downstream face due to the downstream pressure.

Since the force exerted by pressure is the pressure multiplied by the area (�r2

in this case), the force balance becomes

FK � F0 � FL � P0 � � � r2 � PL � � � r2 (7-7)

Simplifying this expression gives

FK � (P0 � PL) � � � r2 � P � � � r2 (7-8)

Combining Eqs. 7-8 and 7-5 yields

(2� � r � L)(1⁄2 � � � Vpore2) � e � P � � � r2 (7-9)

where L is the cylinder length. The friction loss in the cylinder is P. Substitutingin the expression P � � � g � h gives

(2� � r � L)(1⁄2 � � � Vpore2) � e � � � g � h � � � r2 (7-10)


fluid flow

L

Figure 7-6. Cylinder representing element for fluid flow in granular media.

FKP0

F0

PL

FL

Figure 7-7. Force balance on cylinder representing element in granularmedia.

Rearranging to solve for the friction loss in head units, we get

(7-11)

For fixed beds of granular media, the area for flow is not the full, completetubular area. Because the pores through which the fluid flows are not of circularcross section, the hydraulic radius must be used, so r is changed to rH. Substitut-ing for Vpore from Eq. 7-2 gives

(7-12)

The hydraulic radius of the cylinder can be expressed as

(7-13)

Dividing the numerator and denominator by the bed volume gives

(7-14)

where a� is the specific surface area, which is defined as

(7-15)

Substituting Eq. 7-15 into Eq. 7-14 for a� gives an equation for the hydraulicradius for flow through a packed bed of granular media:

(7-16)

Substituting Eq. 7-16 into Eq. 7-12, we have for the head loss equation

(7-17)

By defining a “new” friction factor, f, as 6 � e, Eq. 7-17 becomes

(7-18)h fLd

Vg

��(1 )εε3

02

h eLd

Vg

��

6 02(1 )ε

ε3

r

d

H6

(1 )

�

�

ε

ε

a� �total grain surface area

volume of grains��

�

�

d

dd

2

31

6

6

rH

volume of voids bed volume

wetted surface�

/

area bed volume (1 )/�

�

εεa

�

rHcross section available for flow

wetted p�

eerimeter

volume available for flow

total we�

ttted surface area

h eLr

V

g�

H

0

ε⎛

⎝⎜⎞

⎠⎟

2

h eLr

V

g�

pore2


Grains that make up a packed bed may not be completely spherical, and thesphericity may be expected to affect the frictional loss. The head loss can beadjusted for sphericity of the grains in the bed with a sphericity or shape factor, �:

(7-19)

This is an equation to calculate the clean-bed head loss in granular media andis called the Kozeny-Carman or Ergun equation (see Droste 1997). The Kozeny-Carman friction factor is defined as

(7-20)

where �, the Reynolds number, is quantified with

(7-21)

The Kozeny-Carman friction factor, Eq. 7-20, comprises two terms, a laminarterm and a turbulent term. The laminar term dominates at low Reynolds numbers(approximately �1), and the turbulent term dominates at high Reynolds numbers(approximately �1000). See Fig. 7-8.

� �d V� �0 �

�

f ��

�150 1 751 ε

�

⎛

⎝⎜⎞

⎠⎟.

hf L

dVg

��

�

(1 )εε3

02


Figure 7-8. Dominance of the laminar and turbulent friction factor terms.

Example

Water at 20 °C is passing through a 1.0-m deep sand filter in a 1.5-m-diametercylinder. The filtration rate is 125 L/m2�min and the grain size is 0.50 mm.Assume the porosity is 0.38 and the sphericity is 0.9. What is the head loss?

Solution

Since 125 L/m2�min � 2.083 � 10�3 m/s, the empty bed or superficial velocity is2.083 � 10�3 m/s. At 20 °C, the fluid density is 998.21 kg/m3, and the absolute vis-cosity is 1.002 cP or 1.002 � 10�3 kg/m�s.

Calculating the Reynolds number we have

and so the Kozeny-Carman friction factor is

The Kozeny-Carman equation gives

Thus the friction loss through the sand filter is 1.0 m.

Fluidization of Granular Media

When the head loss or water quality in the effluent from a sand or mixed mediafilter is no longer acceptable, the media is backwashed. In backwashing, the fluidflow is reversed through the bed, and fluid is pumped upward to fluidize andexpand the bed. The high shear forces and grain-to-grain abrasion produced dur-ing fluidization detaches captured particles from the media, allowing the particlesto be removed from the bed with the fluid through the top of the filter.

Fluidization of a bed of granular media is produced as the upflow fluid veloc-ity is increased through the bed until the bed starts to expand. As the velocityincreases, the drag on individual grains increases and grains move apart andbecome suspended in the flow. Once the grains start to separate from each other

hf L

dVg

��

��

�

( ) .

.

( . )

.

1 91 2

0 9

1 0 38

0 38

102

3

εε3

m

0.55 m

m

s

m

s

m

2

�

�

��

�

10

2 083 10

9 81

1 03

3

2

.

.

.

⎛⎝⎜

⎞⎠⎟

f ��

� ��

�150 1 75 1501 04

11 1ε 0.38

�

⎛

⎝⎜⎞

⎠⎟⎛

⎝⎜⎞

⎠⎟.

..775 91 2� .

� � �

� ��

d V� ��

0

3 30 5 10 2 083 10 99�

�

( . ) .mm

s

⎛⎝⎜

⎞⎠⎟

88 21

10

1 043

.

.

kg

m

1.002kg

m s

3

�

��

�


(becoming suspended in the flow), the bed acts like a fluid in that it can bepumped and poured like a fluid, giving rise to the term fluidization.

The condition for fluidization is that the pressure drop through the bed coun-terbalances the weight of the bed per unit cross section. That condition defines theminimum fluidization velocity for the fluid in the bed. For fluidization, the forcebalance on a bed of granular media is shown in Fig. 7-9. The force balance is

F0 � FL � Fg � 0 (7-22)

where Fg is the force of gravity on the bed. F0 � FL is the difference in the pressureforce from the upstream face to the downstream face, which is due to the pressureloss from the fluid flowing through the bed. Force is equal to pressure P multipliedby area A, and we can substitute for the net force of gravity of the bed to get

(7-23)

where �g is the grain density. Because P � �gh and vbed � LA, we have

(7-24)� � �ghAgg

LAg� � �c

( )( )1 ε

ΔP Agg g� � � �

c

bed( )( )1 ε � � v


Media

Fg

FL

F0

P0

PL

Figure 7-9. Force balance on fluidization of granular media.

Dividing by A gives

(7-25)

Equation 7-25, obtained from a force balance, can be solved simultaneouslywith the Kozeny-Carman equation, Eq. 7-19 (with Eqs. 7-20 and 7-21), to find thefluidization velocity, Vfluidization. It is not possible to arrange an equation explicit inVfluidization from these equations, but they may be solved with various techniques/software packages (e.g., MathCad®, Mathematica®, Microsoft Excel®). However,simplifications may be made in the two extreme flow regimes, when completelylaminar or completely turbulent:

(7-26)

(7-27)

In calculating minimum fluidization velocity, the flow regime (� � 1 or � �

1,000) should be chosen or guessed; the fluidization velocity should then be deter-mined by using Eq. 7-26 or 7-27 and verified by calculating the Reynolds numberafter determining the fluidization velocity. If the system is not completely laminaror turbulent, then Eqs. 7-26 and 7-27 cannot be used, and the full set of equationsmust be solved.

The height of the granular media bed remains constant with the grains sup-ported on one another until the minimum fluidization velocity is reached. As thevelocity is increased above the minimum fluidization velocity, the bed height andbed volume increases proportionately as shown in Fig. 7-10.

Example

An 8-m by 3-m sand filter is to be backwashed with water at 20 °C. The sand hasa sphericity of 0.95, a diameter of 0.50 mm, and a grain density of 2.1 g/cm3.Assume the porosity is 0.38. What water flow rate is needed to fluidize themedia?

Solution

At 20 °C, � � 998.21 kg/m3 and � � 1.002 cP � 1.002 � 10�3 kg/m�s.Start by assuming � � 1,000, where we can use Eq. 7-27 to calculate the min-

imum fluidization velocity:

Vdg g

fluidization f��

�

( )

.(

/

ε3

1 75

1 2⎡

⎣⎢⎢

⎤

⎦⎥⎥

oor � �1 000, )

Vg

dgfluidization for�

�

�

( )( )

� �

��

150 112 2ε

ε

3

� �

� � �ghgg

Lg� � �c

( )( )1 ε


Now we have to check to see whether � � 1,000. From Eq. 7-21, we have

Since the Reynolds number is not greater than 1,000, we have to recalculateusing Eq. 7-26.

� � �

� �

d V� ��

fluidization

mm

sñ

ì

5 0 10 0 013 9984. . .221

10

6 33

kg

m

1.002kg

m s

3

��

�

� .

Vflluidization

2m

m

s

kg

m�

� �0 95 5 0 10 9 81 21004. . .� � �33 3

3

kg

m

kg

m

� 998 21 0 38

1 75 998 21

3. ( . )

. .

⎛

⎝⎜⎞

⎠⎟⎡

�

�

⎣⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

1 2

0 013

/

.Vfluidization

m

s�

Vdg g

fluidization ��

�

( )

.

/

ε3

1 75

1 2⎡

⎣⎢⎢

⎤

⎦⎥⎥


bed

heig

htpr

essu

re d

rop

fixedbed

fluidizedbed

fluid velocity

vfluidization

Figure 7-10. Bed height and pressure drop as a function of fluid velocity.

Check Reynolds number:

Which satisfies the requirement that Reynolds number be less than one. SoEq. 7-26 is a valid equation to use for this case.

The required flow rate for fluidization is

A minimum flow rate of 2.0 m3/min of water must be supplied for backwashingthe filter, and the wash troughs of the filter must be able to handle this flow rate.

Symbol List

a� specific surface areaA wetted surface area of cylinder (2�•radius•length)d grain diametere friction factorf Kozeny-Carman friction factorFK force on surface by fluidK kinetic energy per unit fluid volume (1⁄2•�•�pore

2)r cylinder radiusrH hydraulic radius� Reynolds numberQ volumetric fluid flow rateV fluid velocity

Q � � ��1 4 10 8 360

2 03. ( ) .m

sm m

s

1 min

m

min

3

� � �

� � �

� ��

d V� ��

fluidization

mm

sñ

ì

5 0 10 1 4 10 94 3. . 998 21

10

0 723

..

kg

m

1.002kg

m s

3

��

�

�

Vg

d

V

g

fluidization

fludizat

��

�

( )� �

��

1502 2ε

1 ε

3

iion

2 3

m

s

kg

m�

�

�

9 81 2100 998 21

150 1 002

. .

.

⎛

⎝⎜⎞

⎠⎟

� 110

0 38

1 0 380 95 5 0 10

3

32 4 2

�

�

��

kg

m s

m

flui

�

.

.. ( . )

Vddization

m

s� � �1 4 10 3.


V0 superficial velocityvbed bed volumeVpore pore velocityε porosity� shape factor, sphericity� fluid density�g grain density� absolute fluid viscosity

Problems

1. Water at 60 °F (15 °C) flows through a 3.5-ft-deep packed bed of 0.016 in.(0.4-mm) diameter granular media with an empty bed velocity of 0.012 ft/s.Assume a porosity of 0.4 and a sphericity of 0.95. What is the pressure lossin psid?

2. Water at 10 °C is flowing through an ion exchange bed. The ion exchangebeads are 0.6 mm in diameter with a grain density of 1.15 g/cm3 and asphericity of 0.98. The porosity of the bed is 0.38. What is the pressure lossacross the bed? What is the minimum fluidization velocity?

3. A granular media filter that is cylindrically shaped (diameter � 2.5 m, depth �60 cm) is to be backwashed with 18 °C water. The grain diameter is 0.5 mmwith a sphericity of 0.85 and a grain density of 3.8 g/cm3. The porosity of thebed is 0.4. What is the minimum flow rate needed to fluidized the media forbackwashing?

References

Droste, R. L. (1997). Theory and Practice of Water and Wastewater Treatment, Wiley, NewYork, NY.

Tchobanoglous, G., and Schroeder, E. D. (1985). Water Quality, Addison-Wesley, Read-ing, MA.


Documents

Chapter 7